
theorem Th29:
  for L being complete LATTICE, A being Subset of L st A is upper
  holds A in xi L implies A in sigma L
proof
  let L be complete LATTICE, A be Subset of L;
  set T = the Scott TopAugmentation of L;
A1: the RelStr of T = the RelStr of L by YELLOW_9:def 4;
  then reconsider A9=A as Subset of T;
  reconsider A9 as Subset of T;
  assume A is upper;
  then
A2: A9 is upper by A1,WAYBEL_0:25;
  assume A in xi L;
  then A9 is property(S) by A1,Th27,YELLOW12:19;
  then A9 is open by A2,WAYBEL11:15;
  then A9 in the topology of T by PRE_TOPC:def 2;
  hence thesis by YELLOW_9:51;
end;
